There is a worldwide trend toward defined contribution saving plans and growing interest in privatized social security plans. In both environments, individuals are given some responsibility to make their own asset allocation decisions, raising concerns about how well they do at this task. This paper investigates one aspect of the task, namely diversification. We show that many investors have very naive notions about diversification. For example, some investors follow what we call the 1/n strategy: they divide their contributions evenly across the funds offered in the plan. When this strategy (or others only slightly more sophisticated) is used, the assets chosen depend greatly on the make-up of the funds offered in the plan. We find evidence of naive diversification strategies both in experiments using employees at the University of California and the actual behavior of participants in a wide range of savings plans. In particular, we find the proportion of the assets the participants invest in stocks depends strongly on the proportion of stock funds in the plan. The results raise very serious questions about how privatized social security systems should be designed, questions that would be ignored in most economic analyses.